Date: Mar 24, 1995 8:12 PM
Author: DavidBan1@aol.com
Subject: open-ended problems
I do not know if this is an appropriate posting for this list but as a highschool teacher, I am constantly searching for good problems to give to myclasses. What I try to give are open-ended problems that are appropriate forgroup work. I would be glad to throw out two problems that I have used withsuccess this year and wonder if there are others who are interested insharing problems that they have found good. Bothe problems were used in myprecal class but would also be useful for calculus as well.Lifeguard problemA lifeguard (L) at the beach hears a swimmer (S) calling for help 150 yardsdown the beach and 80 yards off shore. She knows she can run on the beachat a rate of 320 yd./min. and swim at a rate of 60 yd./min. She decides toenter the water at a point P which is x yards from the point T on the beachperpendicular to the swimmer.part 1: Solve the problem by determining how far down the beach she shouldrun before starting to swim. (If this is their first exposure to this kindof problem, I will ask several leading questions about how to set the problemup with a diagram, to estasblish what is being minimized and estimatingreasonable values for x and t, generating an equation etc. Equallyinteresting is a discussion of the kind of accuracy that we should beinterested in. We have all had students give answers to questions like thisto the nearest hundredth. Questions I have asked to get at these issues gosomething like this. Identify aspects of this problem that are eitherunrealistic or are only approximations.

How much does it matter if the lifeguard does not start swimming at theoptimum point. Explain your answer with specific examples.

What effect does an error in estimating your swimming speed have on theproblem. How accurate do you think is the estimate of the swimmers distancefrom the shore? The distance of the lifeguard from T?Perhaps the best question to ask would go something like this. Suppose youare in charge of training the lifeguards at a beach. You wish to instructyour lifeguards on how they should decide where to run before they beginswimming when they have to rescue a swimmer. Prepare a clear talk for yourlifeguards. Include a mathematical justification for any advice that youchoose to give your lifeguards

Problem 2:I find that questions of this type are excellent for helping students to seethe relationship between a problem description and a graph. I find ithelpful to do classroom exercises that require students to estimate whatgraphs will look like. These are also good problems to help studentsunderstand the idea of parameters and to estimate the effect that changingparameters will have on a graph. In addition, we can look at symmetry, theeffects that transformations of the original problem.

a) Sketch the graph of the function y = 4 - x^2. Include on the graph thepoint P with coordinates (0, -3).b) Write an equation for the function representing the distance of P from anarbitrary point on the function in part (a) in terms of the x-coordinate ofthe point.c) Use your understanding of the problem and your common sense to sketch acomplete graph of the function in part (b). Be sure to label and scale youraxes. Indicate on your graph the coordinates of any relative maximum pointsor relative minimum points. Check with your calculatord) Is the graph in part (c) symmetric? If so, what symmetry does itexhibit? Explain how you can tell from the equation in part (b) that thefunction is symmetric.e) In part (d) you explained how you could tell the graph was symmetric fromthe equation. Explain how why you could predict this symmetry from the graphand problem description in part (a).f) What are the coordinates of the point on the graph of y = x2 that isclosest to the point (0, 7).g) Compare this problem to the previous problem. Explain.h) Give another problem that is equivalent to the original problem.i) Describe what happens to the distance graph as the point P moves. Include what happens to your graph as P moves along the y-axis. Whathappens if P moves to the right?

I hope these problems spur some interest. I find that as a classroomteacher, I am always trying to find better ways to encourage my students toexplore problems that really require them to develope solutions withsignificant analysis. It is not always easy for me to come up with questionsthat do this effectively. I hope there are others interested in this kind ofexchange.David BannardCollegiate SchoolRichmond, VA Davidban1@aol.com